Mathematics Flashcards

Using Simpson's 1/3 rule with n = 2 intervals to approximate ∫₀² x² dx gives:

Answer: 2.667

Simpson's rule: (h/3)[f(0)+4f(1)+f(2)] = (1/3)[0+4+4] = 8/3 ≈ 2.667, matching the exact value.

What is the rank of the matrix [[1, 2, 3], [4, 5, 6], [7, 8, 9]]?

Answer: 2

Row reduction shows only 2 linearly independent rows (the third is a linear combination of the first two), so rank = 2.

In numerical integration, increasing the number of subintervals generally:

Answer: Decreases the truncation error

More subintervals means smaller step sizes, reducing truncation (approximation) error.

The solution to the linear system Ax = b using Gaussian elimination requires the matrix A to be:

Answer: Non-singular (invertible)

Gaussian elimination finds a unique solution only when A is non-singular (det ≠ 0).

The Poisson distribution is most appropriate for modeling:

Answer: Number of events in a fixed interval of time or space

The Poisson distribution models the count of rare, independent events occurring in a fixed interval.

Stokes' Theorem relates a surface integral of curl to:

Answer: A line integral around the boundary curve

Stokes' Theorem states ∬(∇×F)·dS = ∮F·dr around the bounding curve.

What is the inverse Laplace transform of F(s) = 1/s²?

Answer: t

L⁻¹{1/s²} = t, since L{t} = 1/s².